Calculator Inputs
This implementation solves one-variable nonlinear optimization problems in x. Enter an objective, choose a method, add optional bounds, and define constraints as separate lines.
Example Data Table
| Field | Example Value | Purpose |
|---|---|---|
| Objective function | (x-2)^2 + 0.4*sin(4*x) |
Creates a smooth nonlinear objective with local curvature changes. |
| Method | Newton with line search | Uses first and second derivative estimates for fast local updates. |
| Initial guess | 0.5 | Starting point for the solver iteration path. |
| Bounds | 0 to 4 | Restricts the admissible search region. |
| Inequality constraint | x-3<=0 |
Keeps the solution at or below 3. |
| Penalty factor | 1000 | Strongly discourages infeasible trial points. |
| Tolerance | 0.000001 | Stops the search when step size and gradient become very small. |
Formula Used
Penalized objective: F(x) = f(x) + μ Σ(max(0, gᵢ(x))²) + μ Σ(hⱼ(x)²) + bound penalties
Central difference gradient: F′(x) ≈ [F(x+h) - F(x-h)] / (2h)
Central difference Hessian: F″(x) ≈ [F(x+h) - 2F(x) + F(x-h)] / h²
Gradient descent step: xₖ₊₁ = xₖ - α · d · F′(xₖ)
Newton step: xₖ₊₁ = xₖ - d · F′(xₖ) / F″(xₖ)
Line search idea: the step is repeatedly halved until the penalized objective improves or the attempt becomes too small.
How to Use This Calculator
- Enter a nonlinear objective in terms of
x. - Select Newton or gradient descent according to your preference.
- Provide an initial guess close to a likely optimum.
- Add optional lower and upper bounds for the search region.
- Enter inequality constraints with
<=or>=, one per line. - Enter equality constraints with
=, one per line. - Set the learning rate, damping, tolerance, penalty factor, and derivative step size.
- Click Solve Optimization Problem to view summary metrics, iteration history, and the Plotly graph above the form.
Frequently Asked Questions
1. What type of optimization does this solver handle?
It handles one-variable nonlinear optimization with optional bounds, inequality constraints, and equality constraints. The solver converts constraints into penalties and iteratively reduces the penalized objective.
2. When should I choose Newton instead of gradient descent?
Choose Newton when the objective is smooth and the starting point is reasonable. Choose gradient descent when curvature estimates are unstable or when you want a simpler update rule.
3. Why does the calculator use a penalty factor?
The penalty factor discourages infeasible solutions. Larger values punish violations more strongly, pushing the search toward points that satisfy bounds and constraints better.
4. What does the gradient value tell me?
A gradient near zero suggests the penalized objective is flattening locally. That often indicates convergence, though the final point can still depend on starting values and constraints.
5. Why might two starting guesses produce different answers?
Nonlinear objectives may contain multiple local minima. Different starting guesses can lead the solver toward different attractive regions, especially when the function is highly oscillatory.
6. What does the line search do?
Line search reduces an overly aggressive step until the penalized objective improves. It helps stabilize the method when gradients are steep or curvature changes quickly.
7. Can I enter expressions like sin, exp, sqrt, and logs?
Yes. The calculator supports sin, cos, tan, asin, acos, atan, sqrt, log, ln, exp, abs, floor, ceil, and round, along with constants pi and e.
8. What do the CSV and PDF buttons export?
They export the result summary and full iteration history. This makes it easier to document convergence, compare configurations, or share solver outcomes in reports.